By [1]}, this operator commutes with an operator of order \(4q+2\)  with \(q\ge g=2\) . We fixed \(4q+2=18\) , that is \(q=4\) , and computed an operator \(M\)  of order 18 in the centralizer \(\mathcal {C}(L_4)\) . We used a Grünbaum's style approach, forcing \([L_4,M_{18}]=0\)  for an arbitrary operator \(M_{18}\)  of order 18. From the family of operators obtained we chose
\(M=\partial ^{18}+9\big (x^4+1\big )\partial ^{16}+288x^3\partial ^{15}+\big (36x^8+72 x^4+4572 x^2+15\big )\partial ^{14}\\\hphantom{M=}{}+H_14\partial ^{14}+\cdots +H_0\) 
